C₂×#147125 #72309795

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁷

Aliases

direct product

C₂×#147125

Subgroups

	subgroup	G/H
center	C₂³×C₄	C₂×C₄²
commutator	C₂²×C₄	C₂²×C₄²
Frattini, cosocle	C₂³×C₈	C₂⁴
socle	C₂⁴	C₂×#97
prime radical	C₂⁵	C₂×M₁₆
LCS	G ▷ C₂²×C₄ ▷ C₁
UCS	C₁ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

3489 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 26
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 26 points:

(3,4)(5,6)(7,8,9,10);(3,4)(5,6)(8,10)(12,16,17,19)(13,20)(18,26,25,24)(21,23);(1,2);(4,6)(11,12,13,18,14,19,21,26,15,17,20,25,22,16,23,24)

Conjugacy classes & character table

208 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72256927, C₂×#72250110, C₂×#72239256
maximal normal subgroup of	C₂×#72256927, C₂×#72250110, C₂×#72239256
maximal quotient of	#72323009

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:

gap> G := Group((3,4)(5,6)(7,8,9,10), (3,4)(5,6)(8,10)(12,16,17,19)(13,20)(18,26,25,24)(21,23), (1,2), (4,6)(11,12,13,18,14,19,21,26,15,17,20,25,22,16,23,24));

C2×#147125 #72309795